Is the world mathematical?

One frequently encounters the question Why is the world (or nature) mathematical? (What is generally intended is Why does nature obey mathematical laws?, or Why does mathematics describe nature so well?) The jumping-off place for this type of question is the idea that mathematics has been shown to be effective in describing some important aspects of the world, particularly in the natural sciences. (The Unreasonable Effectiveness of Mathematics in the Natural Sciences, by Eugene Wigner)
But Why is the world mathematical? to me begs the question: IS the world mathematical? […]

So the confusion about Why is mathematics effective? may lie in the confusion that follows when one makes a distinction or separation between the world and mathematics. The stuff of the world IS mathematics.

Eugene Wigner wrote a famous essay on the unreasonable effectiveness of mathematics in natural sciences. He meant physics, of course. There is only one thing which is more unreasonable than the unreasonable effectiveness of mathematics in physics, and this is the unreasonable ineffectiveness of mathematics in biology.

Besides being one of the most influential mathematicians (and mathematical physicists) of 20th century, Gelfand also has 50 years of experience of research in molecular biology and biomathematics, and his remark deserves some attention.

Indeed biology, and especially molecular biology, is not a natural science in the same sense as physics. Indeed, it does not study the relatively simple laws of the world. Instead, it has to deal with molecular algorithms (such as, say, the transcription of RNA and synthesis of proteins which ensures the correct spatial shape of the protein molecule) which were developed in the course of evolution as a way of adapting living organisms to the changing world. If they solve a particular problem in an optimal way, they should allow some external description in terms of the structure of the problem. Indeed, this is the principal paradigm of physics; it is an experimental fact that the behavior of physical systems is governed by various minimality / maximality principles, and the optimal points have, as a rule, especially nice mathematical properties.

But why should a biological system to be globally optimal? Evolution is blind, and there is no reason to assume that the optimal solution is reached. The implemented solution could be one of myriads of local optima, sufficiently good to ensure survival. Lucky strikes could be so rare that the huge search space and billions of years of evolution produced just one survivable algorithm, which, as a result, dominates the living world, and is perceived by us as something special. But it might happen that there is absolutely no external characterization which allows us to distinguish it from other possible solutions, and that its evolutionary phylogeny is its only explanation.

However, I am not a philosopher and cannot claim that my solution of Gelfand’s paradox is correct. What I claim is that philosophers ask wrong questions. The classical conundrum of relations between mathematics and physical world starts to look very different — and much more exciting — as soon as we include biology into consideration. I will try to continue this discussion.

Responses

I don’t want to say anything stupid, but I can’t help but think (and have thought) that this question “Why is mathematics effective?” is sort of a blunder. I do not know much about the platonism vs realism debate, but I have an inkling that my fallacy may lie in that argument. I also have not read the “Why is mathematics effective?” link.

If you still care to read, the blunder that I think I see has to do with my opinion that Mathematics is a language like any other languages; its effective within a particular domain of thought and is just as effective as we can use it (like “normal language”).

While, and this may be overly simplistic, “normal language” tends to convey qualities (domain of quality), it seems mathematics covers quantities (domain of quantity). Physics can be qualitatively done well with normal language, and few people seem to consider this (as far as I can tell). That is, there is a better question to ask than “Why is mathematics effective?” I think the bigger, better, more general and oh-so-fun question is, why is language so effective? But maybe even more-so, why is nature so readily available to being defined in such a way that it obeys logic? Which is just another way of asking “Why is nature so consistent?”

One problem with Wigner’s argument is that for many physical domains, our only means of apprehending the domain is via the mathematical model itself. How could we possibly tell, for example, whether or not String Theory is effective in describing the world? We have no independent means of accessing the domain which String Theory purports to describe in order to compare with String Theory. And perhaps, since String Theory purports to describe dimensions inaccessible to us, we never will have.

And, for those physical domains where we do have independent means to assess our mathematical models, it is not surprising to me that the mathematical models fits their respective domains well. They were designed to! Wigner’s argument is akin to saying, “Isn’t it wonderful that we humans have two legs! Because that means we can use all the pairs of trousers that have been made over the years.”

Sasha Do you think that if mathematicians were more involved in applications to biology (and maybe economics) that would result in new kinds of mathematics that are “surprisingly effective” in those areas? As you say problems in biology (and I suppose economics) can be formulated as minimization of some functional, but the processes are stochastic not deterministic. But is this not rather like non-equilibrium thermodynamics?

@Bill : I do not know for Alexander, but my own opinion is that there are new kinds of mathematics to think and develop from biology, and from psychology at least.
I am not so confident about economics.

I do not know if they will be surprisingly effective but I do hope we will find them surprisingly obvious in retrospect and that it will generate endless a posteriori comments by pseudo-philosophers of science.

I do not know if our generation will succeed in starting these new branches of mathematics, nor do we know what were the embryonic stages and the timeframe of formation of the discrete and continuous concepts in humanoid culture and what were the real difficulties to overcome (for instance what was the dependency with language is not clear).

[M]y own opinion is that there are new kinds of mathematics to think and develop from biology, and from psychology at least. I am not so confident about economics.

I do not know if they will be surprisingly effective but I do hope we will find them surprisingly obvious in retrospect and that it will generate endless a posteriori comments by pseudo-philosophers of science.

Largely, this is an opinion expressed in [1] [2] by Gromov. You may enjoy his remarks
about “Sturtevant’s structure recognition paradigm that has not been absorbed yet
by the present day mathematics”, and the paper of Mendel [3]